Paper 4, Section II, C

Dynamics and Relativity | Part IA, 2021

A particle of mass mm follows a one-dimensional trajectory x(t)x(t) in the presence of a variable force F(x,t)F(x, t). Write down an expression for the work done by this force as the particle moves from x(ta)=ax\left(t_{a}\right)=a to x(tb)=bx\left(t_{b}\right)=b. Assuming that this is the only force acting on the particle, show that the work done by the force is equal to the change in the particle's kinetic energy.

What does it mean if a force is said to be conservative?

A particle moves in a force field given by

F(x)={F0ex/λx0F0ex/λx<0F(x)=\left\{\begin{array}{cc} -F_{0} e^{-x / \lambda} & x \geqslant 0 \\ F_{0} e^{x / \lambda} & x<0 \end{array}\right.

where F0F_{0} and λ\lambda are positive constants. The particle starts at the origin x=0x=0 with initial velocity v0>0v_{0}>0. Show that, as the particle's position increases from x=0x=0 to larger x>0x>0, the particle's velocity vv at position xx is given by

v(x)=v02+ve2(ex/λ1)v(x)=\sqrt{v_{0}^{2}+v_{e}^{2}\left(e^{-|x| / \lambda}-1\right)}

where you should determine vev_{e}. What determines whether the particle will escape to infinity or oscillate about the origin? Sketch v(x)v(x) versus xx for each of these cases, carefully identifying any significant velocities or positions.

In the case of oscillatory motion, find the period of oscillation in terms of v0,vev_{0}, v_{e}, and λ\lambda. [Hint: You may use the fact that

w1duuuw=2cos1ww\int_{w}^{1} \frac{d u}{u \sqrt{u-w}}=\frac{2 \cos ^{-1} \sqrt{w}}{\sqrt{w}}

for 0<w<10<w<1.]

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